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Data and Curse of Dimensionality; Spectral Geometry comes to rescue!

High dimensional data is increasingly available in many fields. However, the analysis of such data
suffers the so-called curse of dimensionality. One powerful approach to nonlinear dimensionality
reduction is the diffusion-type maps. Its continuous counterpart is the embedding of a manifold
using the eigenfunctions of the Laplace-Beltrami operator. Accordingly, one may ask, how many
eigenfunctions are required in order to embed a given manifold.

In this talk, I will give some background regarding the dimensionality reduction problem,
spectral geometry, and show theoretical results for a generalized diffusion map. Specifically, I will
show a closed Riemannian manifold can be embedded into a finite dimensional Euclidean space
by maps constructed based on the connection Laplacian at a certain time. This time and the
embedding dimension can be bounded in terms of the dimension and geometric bounds of the
manifold. Furthermore, the map based on heat kernels can be made arbitrarily close to an
isometry. In addition, I will give a ‘’real world” example pertaining to paleonthology, that
demonstrates how heat kernels and diffusion maps can be used to quantify the similarity of
shapes. The empirical results suggest that this framework is better than the metric commonly
used in biological morphometrics.